Iterative detection in MIMO systems

ABSTRACT

Detection for a MIMO (multiple-input, multiple-output) wireless communications system with symbols iteratively detected in subsets with maximum likelihood hard decisions within subsets. Previously detected subsets of symbols are used to regenerate corresronding input signals for interference cancellation. With a 4-transmitter antenna, 4-receiver antenna system, two subsets of two symbols are possible with the first two symbols detected with zero-forcing or MMSE soft estimates which feed maximum likelihood hard decisions; and the hard decision for the first two symbols are used for interference cancellation followed by zero-forcing or MMSE soft estimates for the second two symbols which then feed further maximum likelihood hard decisions.

CROSS-REFERENCE TO RELATED APPLICATIONS

Priority is claimed from provisional applications: Appl.Nos. 60/395,604,filed Jul. 12, 2002, and 60/399,746, filed Jul. 30, 2002. The followingcopending application dislcoses related subject matter and has a commonassignee: application Ser. No. 10/447,978, filed May 29, 2003.

BACKGROUND OF THE INVENTION

The present invention relates to communication systems, and moreparticularly to multiple-input multiple-output wireless systems.

Wireless communication systems typically use band-limited channels withtime-varying (unknown) distortion and may have multi-users (such asmultiple cellphone users within a cell). This leads to intersymbolinterference and multi-user interference, and requiresinterference-resistant detection for systems which are interferencelimited. Interference-limited systems include multi-antenna systems withmulti-stream or space-time coding which have spatial interference,multi-tone systems, TDMA systems having frequency selective channelswith long impulse responses leading to intersymbol interference, CDMAsystems with multi-user interference arising from loss of orthogonalityof spreading codes, high data rate CDMA which in addition to multi-userinterference also has intersymbol interference.

Interference-resistant detectors commonly invoke one of three types ofequalization to combat the interference: maximum likelihood sequenceestimation, (adaptive) linear filtering, and decision-feedbackequalization. However, maximum likelihood sequence estimation hasproblems including impractically large computation complexity forsystems with multiple transmit antennas and multiple receive antennas.Linear filtering equalization, such as linear zero-forcing and linearminimum square error equalization, has low computational complexity buthas relatively poor performance due to excessive noise enhancement. Anddecision-feedback (iterative) detectors, such as iterative zero-forcingand iterative minimum mean square error, have moderate computationalcomplexity but only moderate performance.

Thus there is a problem of trade-off of performance and computationalcomplexity for multiple-input, multiple-output systems.

SUMMARY OF THE INVENTION

The present invention provides a method and detector for multiple-input,multiple-output (MIMO) systems which iteratively detects subsets ofsymbols and thereby allows reduced-state maximum likelihood decisions.

This has advantages including improved performance with moderateincrease in computational complexity.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1 a-1 b are flow and functional block diagrams.

FIGS. 2 a-2 c illustrate receivers and transmitters.

FIGS. 3 a-3 d are simulation results.

FIGS. 4 a-4 b show Turbo coding.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

1. Overview

Preferred embodiment detectors and detection methods for multi-input,multi-output (MIMO) systems partition the set of transmitted symbolsinto subsets and jointly detect the symbols within a subset anditeratively detect the subsets. This provides interference cancellationof previously detected symbols. The joint detection within a subset maybe a linear soft detection followed by maximum likelihood (ML) harddecision. FIG. 1 a is a flow diagram, and FIG. 1 b illustratesfunctional blocks of a preferred embodiment which partitions of a set ofP symbol indices, {1, 2, . . . , P}, into M subsets, I₁, I₂, . . . ,I_(M), of sizes P₁, P₂, . . . , P_(m), respectively. Thus,P=Σ_(1≦m≦M)P_(m). The hard estimates for the P_(m) symbols with indicesin subset I_(m) are computed by a P_(m)-input, P_(m)-output ML detector.These hard estimates are then used to regenerate the signal componentsassociated with the symbols whose indices lie in I_(m), and theregenerated components from this iteration plus prior iterations aresubtracted from the received signal (interference cancellation) for useas input to the next iteration. This use of ML detection in reducedstates (subsets of indices) provides a range of performance-complexitytrade-offs previously unavailable.

Preferred embodiment wireless communication systems incorporatepreferred embodiment detection methods. FIG. 1 b illustrates functionalblocks of a preferred embodiment detector. FIG. 2 a shows acorresponding receiver for a MIMO wireless communications system; andFIG. 2 b shows a P-antenna transmitter for such a system. In suchsystems components, base stations, and mobile users, could each includeone or more application specific integrated circuits (ASICs),(programmable) digital signal processors (DSPs), and/or otherprogrammable devices with stored programs for implementation of thepreferred embodiment. The base stations and mobile users may alsocontain analog integrated circuits for amplification of inputs to oroutputs from antennas and conversion between analog and digital; andthese analog and processor circuits may be integrated on a single die.The stored programs may, for example, be in external or onboard ROM,flash EEPROM, and/or FeRAM. The antennas may be parts of RAKE detectorswith multiple fingers for each user's signals. The DSP core could be aTMS320C6xxx or TMS320C5xxx from Texas Instruments.

2. Iterative Detectors

First consider iterative detectors generally. FIG. 2 a illustrates areceiver with an interference-resistant detector as could be used in awireless communications system with P transmit antennas (P data streams)and Q receive antennas. FIG. 2 b illustrates a corresponding transmitterwith P transmit antennas. The received signal in such a system can bewritten as:r=Hs+wwhere r is the Q-vector of samples of the received baseband signal(complex numbers) corresponding to a transmission time n:

${r = \begin{bmatrix}{r_{1}(n)} \\{r_{2}(n)} \\\vdots \\{r_{Q}(n)}\end{bmatrix}};$s is the P-vector of transmitted symbols (complex numbers of a symbolconstellation) for time n:

${s = \begin{bmatrix}{s_{1}(n)} \\{s_{2}(n)} \\\vdots \\{s_{P}(n)}\end{bmatrix}};$H is the Q×P channel matrix of attenuations and phase shifts; and w is aQ-vector of samples of received (white) noise. That is, the (q,p)thelement of H is the channel (including multipath combining andequalization) from the pth transmit source to the qth receive sink, andthe qth element of w is the noise seen at the qth receive sink.

Note that the foregoing relation applies generally to various systemswith various interference problems and in which n, r, s, P, and Q havecorresponding interpretations. For example:

-   (a) High data rate multi-antenna systems such as BLAST (Bell Labs    layered space time) or MIMO and multi-stream space-time coding:    spatial interference suppression techniques are used in detection.-   (b) Broadband wireless systems employing OFDM (orthogonal frequency    division multiplex) signaling and MIMO techniques for each tone or    across tones.-   (c) TDMA (time division multiple access) systems having    frequency-selective channels with long impulse response which causes    severe ISI (intersymbol interference). Use equalizers to mitigate    ISI.-   (d) CDMA (code division multiple access) systems having    frequency-selective channels which cause MUI (multi-user    interference) as a result of the loss of orthogonality between    spreading codes. For high data rate CDMA systems such as HSDPA and    1xEV-DV, this problem is more severe due to the presence of ISI.    Equalizers and/or interference cancellation may be used to mitigate    these impairments.-   (e) Combinations of foregoing.

Hence, P is essentially the number of symbols that are jointly detectedas they interfere with one another, and Q is simply the number ofcollected samples at the receiver. Because there are P independentsources, Q must be at least as large as P to separate the P symbols. A(interference-resistant) detector in a receiver as in FIGS. 2 a-2 banalyzes received signal vector r and outputs soft estimates x of thetransmitted symbols s to a demodulator and decoder.

Presume that the different symbols which are transmitted via P differentantennas are uncorrelated and also may utilize different modulationschemes. This implies the P×P matrix of expected symbol correlations,Λ=E[ss^(H)], is diagonal with entries equal the expected symbol energies(λ_(k)=E[|S_(k)|²]):

$\Lambda = \begin{bmatrix}\lambda_{1} & 0 & \cdots & 0 \\0 & \lambda_{2} & \cdots & 0 \\\vdots & \vdots & ⋰ & \vdots \\0 & 0 & \cdots & \lambda_{P}\end{bmatrix}$

For linear filtering equalization detectors, such as linear zero-forcing(LZF) or linear minimum mean square error (LMMSE), the soft estimates,denoted by P-vector x, derive from the received signal by linearfiltering with P×Q matrix F; namely, x=Fr.

In particular, LMMSE detection finds the matrix F by minimizing the meansquare error, E[∥x−s∥²]. With perfect estimation of the channel H, theminimizing matrix F is given by:

$\begin{matrix}{F_{LMMSE} = {\lbrack {{H^{H}H} + {\sigma^{2}\Lambda^{- 1}}} \rbrack^{- 1}H^{H}}} \\{= {\Lambda\;{H^{H}\lbrack {{H\;\Lambda\; H^{H}} + {\sigma^{2}I_{Q}}} \rbrack}^{- 1}}}\end{matrix}$where σ² is the variance per symbol of the additive white noise w andI_(Q) is the Q×Q identity matrix. Note F has the form of a product of anequalization matrix with H^(H) which is the matrix of the matched filterfor the channel.

Analogously, LZF detection finds the matrix F by inversion:F _(LZF) =[H ^(H) H] ⁻¹ H ^(H)

Essentially, LZF detection nulls out the interference at the expense ofexcessive noise enhancement, while LMMSE detection minimizes the sum ofinterference and noise energy to provide a balanced trade-off betweenbias (due to residual interference) and variance (due to noiseenhancement). But then each component of x is processed separately toobtain a corresponding hard symbol estimate. That is, a hard estimate,ŝ_(i), for the ith symbol, s_(i), comes from application of a harddecision operator to the soft estimate: ŝ_(i)=

{x_(i)} for i=1, 2, . . . , P, where

{} is a non-linear decision device, usually a single-input,single-output (SISO) maximum-likelihood (ML) detector.

A typical iterative (decision-feedback) detector (IZF or IMMSE) forblocks of P symbols has a series of P linear detectors (P iterations)with each linear detector followed by a (hard) decision device andinterference subtraction (cancellation). Each of the P linear detectors(iterations) generates both a hard and a soft estimate for one of the Psymbols. The hard estimate is used to regenerate the interferencearising from the already-estimated symbols which is then subtracted fromthe received signal, and the difference used for the next linear symbolestimation. Hence, the hard symbol estimates, ŝ_(i), are also computedin a disjoint fashion.

On the other hand, the optimal joint maximum likelihood (ML) detectorgenerates the hard symbol estimates as:ŝ _(ML)=arg min_(sεC) ∥r−Hs∥ ²where C=C₁×C₂× . . . ×C_(P) is the P-dimensional constellation ofP-vectors of symbols and has size Π_(1≦j≦P) |C_(j)| where |C_(j)|denotes the number of elements in constellation C_(j). Thus the hardsymbol estimates are jointly computed, and the computational complexityof this optimal detection is proportional to the constellation sizewhich is exponential in P.

When an error correcting code is employed, soft symbol or soft bit (loglikelihood ratio) estimates are often desired (see Section 8 forexample). While soft symbol estimates can be obtained in many ways, theconditional expectation estimator is optimal. The conditionalexpectation estimator is:

$\hat{s} = {{E\lbrack {s❘r} \rbrack} = \frac{\sum\limits_{s \in C}{s \times {\exp( {{- {{r - {Hs}}}^{2}}/\sigma^{2}} )}}}{\sum\limits_{s \in C}{\exp( {{- {{r - {Hs}}}^{2}}/\sigma^{2}} )}}}$The performance loss of iterative detectors comes from the disjointprocessing of the soft symbol estimates, x_(i), to obtain thecorresponding hard symbol estimates, ŝ_(i). The preferred embodimentdetectors circumvent this problem by partial ML detections with moderatecomputational complexity.3. Iterative Reduced-state Maximum-likelihood Detection

FIG. 1 a is a flow diagram and FIG. 1 b shows functional blocks forfirst preferred embodiment detection methods which detect a P-vector oftransmitted symbols, s, from a received Q-vector signal, r, as expressedin the following pseudocode which presumes a partition of the set ofsymbol indices, {1, 2, . . . , P}, into M subsets of indices I₁, I₂, . .. , I_(M), where I_(m) has P_(m) elements.

initialize r⁽¹⁾ = r for m = 1 to M x^((m)) = F_(m)r^((m)) //P_(m)-vector of soft estimates for I_(m) indexed symbols ŝ^((m)) =MLD_(m){x^((m))} // P_(m)-vector of hard decisions for I_(m) indexedsymbols r^((m+1)) = r^((m)) − G_(m)ŝ^((m)) // interference cancellationusing hard estimates endwhere F_(m) is a P_(m)×Q matrix for the feedforward transformation ofreceived signal into soft estimates, x^((m)), of the I_(m) indexedsymbols; MLD_(m) is a (P_(m), P_(m)) maximum likelihood (ML) detector;G_(m) is a Q×P_(m) matrix for the feedback transformation using the mthiteration output for interference cancellation. The following paragraphsdescribe choices for F_(m), G_(m), and MLD_(m). And note that thepreferred embodiment methods for trivial index set partitions reduce tothe usual methods: for M=1 (no partitioning), the method is just an MLdetection for all of the symbols, and for M=P (every index is a separatesubset), the method is just regular iterative detection.

Zero-forcing and minimum mean square error criteria each define F_(m)and G_(m) as follows. Begin with some convenient notation: let J₁={1, 2,. . . , P} and recursively take J_(m)=J_(m−1)−I_(m−1) for 2≦m≦M, soJ_(m) is the set of indices of symbols remaining to be detected at thestart of the mth iteration. Let H_(m) be the Q×P_(m) matrix consistingof the columns of H with index in I_(m), so H_(m) is the channel matrixfor the P_(m) symbols being detected during the mth iteration. Let K_(m)be the Q×(Σ_(m≦k≦M) P_(k)) matrix consisting of the columns of H withindex in J_(m), so K_(m) is the channel matrix for the symbols remainingto be detected at the start of the mth iteration (K₁=H). Let Λ_(m) bethe P_(m)×P_(m) diagonal sub-matrix of Λ of elements with both row andcolumn indices in I_(m); consequently, the elements of Λ_(m) are theenergies of the symbols to be detected during the mth iteration. Let

_(m) be the (Σ_(m≦k≦M) P_(k))×(Σ_(m≦k≦M) P_(k)) diagonal sub-matrix of Λof elements with both row and column indices in J_(m), so the elementsof

_(m) are the energies of the symbols remaining to be detected at thestart of the mth iteration. Let 1_(m) be the P_(m)×Q the matrix of therows of the identity matrix I_(Q) with row index in I_(m); that is,1_(m(p,q))=1 if q is the pth element of I_(m) and =0 otherwise, and thus1_(m) extracts the subspace spanned by the symbols to be detected in themth iteration.

Neglecting decision feedback error and choosing F_(m) and G_(m)according to the zero-forcing criterion yields for m=1, 2, . . . , M−1:F_(m)=1_(m)[K_(m) ^(H)K_(m)]⁻¹K_(m) ^(H)G_(m)=H_(m)

Analogously, choosing F_(m) and G_(m) according to the minimum meansquare error criterion yields for m=1, 2, . . . , M−1:

$\begin{matrix}{F_{m} = {{1_{m}\lbrack {{K_{m}^{H}K_{m}} + {\sigma^{2}\Pi_{m}^{- 1}}} \rbrack}^{- 1}K_{m}^{H}}} \\{= {\Lambda_{m}{H_{m}^{H}\lbrack {{K_{m}\Pi_{m}\; K_{m}^{H}} + {\sigma^{2}I_{Q}}} \rbrack}^{- 1}}}\end{matrix}$ G_(m) = H_(m)In both cases during the last iteration (m=M) all remaining symbols willbe detected, so J_(M)=I_(M), K_(M)=H_(M),

_(M)=Λ_(M). Further, there is no next iteration and thus no need for acancellation matrix G_(M); see FIG. 1 b. The following sections 4 and 5provide an explicit example and an extension to compensate for decisionfeedback error.

The maximum likelihood hard decision for the mth iteration, MLD_(m),depends upon the signal model for r^((m)). In particular,

$\begin{matrix}{r^{(m)} = {r^{({m - 1})} - {G_{m - 1}{\hat{s}}^{({m - 1})}}}} \\{= {r - {\sum\limits_{1 \leq k \leq {m - 1}}{G_{k\;}{\hat{s}}^{(k)}}}}} \\{= {{H\; s} + w - {\sum\limits_{1 \leq k \leq {m - 1}}{G_{k\;}{\hat{s}}^{(k)}}}}}\end{matrix}$where ŝ^((k)) is the P_(k)-vector of hard estimates of s^((k)) which isthe P_(k)-vector of symbols with indices in subset I_(k).

Now for the case of no feedback error, G_(k)=H_(k) and ŝ^((k))=s^((k)),so the model becomesr ^((m))=Σ_(m≦j≦M) H _(j) s ^((j)) +wThus the soft estimates can be written as:

$\begin{matrix}{x^{(m)} = {F_{m}r^{(m)}}} \\{= {{F_{m}H_{m}s^{(m)}} + {\sum\limits_{{m + 1} \leq j \leq M}{F_{m}H_{j}s^{(j)}}} + {F_{m}w}}} \\{= {{F_{m}H_{m}s^{(m)}} + i_{m}}}\end{matrix}$where i_(m) is the residual interference-plus-noise term correspondingto the mth iteration detection. Note that the desired signal term,F_(m)H_(m)s^((m)), is uncorrelated to the residualinterference-plus-noise and thatE[i _(m) i _(m) ^(H) ]=F _(m) K _(m+1)

_(m)(F _(m) K _(m+1))^(H)+σ² F _(m) F _(m) ^(H)For notational convenience define ψ_(m)=E[i_(m)i_(m) ^(H)] which is aP_(m)×P_(m) matrix. Then assuming the residual interference isapproximately Gaussian with covariance matrix ψ_(m), so log-likelihoodmetric for MLD_(m) is just the Euclidean metric scaled by the covariancematrix:ŝ ^((m))=arg min_(c(m))∥Ψ_(m) ^(−1/2)(x ^((m)) −F _(m) H _(m) c^((m)))∥²where c^((m)) is a P_(m)-vector of symbols in the product of the P_(m)constellations of symbols with indices in I_(m). Thus the hard decisionis found by computing all of the metrics and taking the symbol vectorwhich generates the minimum metric; and if all of the constellationshave the same size |C| (such as all are |C|−QAM), then the number ofmetrics to compute is |C|^(Pm).

Note that for F_(m) and G_(m) defined using the zero-forcing criterionthere is no residual interference because F_(m)K_(m+1)=0. In this casei_(m) is exactly Gaussian. For the MMSE criterion, however, someresidual interference exists; hence, i_(m) is a mixture of Gaussian(from the noise) and non-Gaussian (residual interference) randomprocesses. For sufficiently dense symbol constellations and/or asufficiently large number of antennas, the interference can be wellapproximated as Gaussian. Otherwise, a variation of the MLD decision forMMSE-based detection is the use of the exact probability densityfunction (pdf) of i_(m) in the maximum likelihood detection. Becausei_(m) is the sum of F_(m) w and Σ_(m+1≦j≦M) F_(m) H_(j) s^((j)), its pdfis the convolution of their pdfs.

4. 4×4 Example

As an example consider the 4×4 MIMO system with the preferred embodimentpartitioning the symbols into two subsets of two symbols each; that is,P=Q=4 with P₁=P₂=2 (M=2). Explicitly, take I₁={1,2} and I₂={3,4}; thenwith h_(k) denoting the kth column of 4×4 channel matrix H, thezero-forcing criterion yields:F₁=1_({1,2})[H^(H)H]⁻¹H^(H)  (2×4 matrix)G₁=[h₁h₂]  (4×2 matrix)F₂=[h₃h₄]^(H)  (2×4 matrix)Analogously, the MMSE criterion gives:

$\mspace{11mu}\begin{matrix}\begin{matrix}{F_{1} = {{1_{\{{1,2}\}}\lbrack {{H^{H}H} + {\sigma^{2}\Lambda^{- 1}}} \rbrack}^{- 1}H^{H}}} \\{= {\begin{bmatrix}{\lambda_{1}h_{1}^{H}} \\{\lambda_{2}h_{2}^{H}}\end{bmatrix}\lbrack {{\lambda_{1}h_{1}h_{1}^{H}} + {\lambda_{2}h_{2}h_{2}^{H}} + {\sigma^{2}I_{4}}} \rbrack}^{- 1}}\end{matrix} & ( {2 \times 4\mspace{14mu}{matrix}} ) \\{G_{1} = \lbrack {h_{1}h_{2}} \rbrack} & ( {4 \times 2\mspace{14mu}{matrix}} ) \\{F_{2} = \lbrack {h_{3}h_{4}} \rbrack^{H}} & ( {2 \times 4\mspace{14mu}{matrix}} )\end{matrix}$The simulations of section 9 compare performance of variousmodifications of this 4×4 MIMO system preferred embodiment plus regulariterative detection.5. Feedback Error Compensation

In general, the error-free decision feedback assumption is sufficientlyaccurate for high signal-to-noise ratio (SNR) and good channelconditions. Otherwise, error propagation may occur. When the MMSEcriterion is used, the effect of error propagation can be taken intoaccount by using the second order model which presumes the correlationof the transmitted symbols and their hard estimates is expressed by adiagonal P×P energy-correlation matrix:

${E\lbrack {s\mspace{11mu}{\hat{s}}^{H}} \rbrack} = \begin{bmatrix}{\lambda_{1}\rho_{1}} & 0 & \cdots & 0 \\0 & {\lambda_{2}\rho_{2}} & \cdots & 0 \\\vdots & \vdots & ⋰ & \vdots \\0 & 0 & \cdots & {\lambda_{P}\rho_{P}}\end{bmatrix}$where the symbol-hard estimate correlations ρ_(j) are generallyreal-valued and in the range −1≦ρ_(j)≦+1. Let R denote the P×P diagonalmatrix of these correlations. Then define the P_(m)×P_(m) diagonalcorrelation submatrix R_(m) with diagonal elements ρ_(k) for k in I_(m)and the (Σ_(m≦k≦M)P_(k))×(Σ_(m≦k≦M)P_(k)) diagonal correlation submatrix

_(m) with diagonal elements ρ_(k) for k in J_(m). Thus, R_(m) is usefulfor interference cancellation at the end of the mth iteration when theŝ^((m)) are available, so the ρ_(k) for k in I_(m) are generallynon-zero in R_(m). Further,

_(m) is useful for interference suppression at the start of the mthiteration when the ŝ^((m)) are not yet available and so the ρ_(k) for kin I_(m) are set to zero in forming

_(m). All the ρ_(k) for k in J_(m) are not yet available so this is all0s. For multistage detection, some ρ_(k) will be 0 for the undetectedsymbols in the first stage; during the second stage estimates from theprevious stage for all the symbols are available and ρ_(k) is non-zerofor every k.

Taking into account the effect of error propagation in the MMSEcriterion approach leads to modified F_(m) and G_(m); for m=1, 2, . . .M−1:

F m = 1 m ⁡ [ K m H ⁢ K m ⁡ ( I - m 2 ) + σ 2 ⁢ m - 1 ] - 1 ⁢ K m H = Λ m ⁢ Hm H ⁡ [ K m ⁡ ( I - m 2 ) ⁢ m ⁢ K m H + σ 2 ⁢ I Q ] - 1 G m = H m ⁢ R mNote that these modified matrices reduce to the prior expressions whenρ_(k)=1 for previously-detected symbols. In particular, ρ_(k)=1 for k inJ_(m)−I_(m) reduces F_(m) to the prior expression and ρ_(k)=1 for k inI_(m) reduces G_(m) to the prior expression.

This feedback error compensation by use of correlations to limitcancellation allows for a multistage detection with each stage a set ofM iterations as in the foregoing and with the hard estimates andcorrelations of one stage used as initial conditions for the next stage.In this manner, performance can be improved in stages by controlling thecancellation according to current correlation. Of course, thecorrelations must be computed, either by ensemble averaging (i.e., use apilot channel or a set of training symbols that undergo the same MIMOchannel as the information symbols) or by use of an analytical model(e.g., specific symbol constellation and Gaussian approximations) whichcan relate correlations and SINRs (which are used to define the F and Gmatrices) plus provide convergence to limiting values through successivestages.

In particular, if superscript/subscript [n] denotes nth stage variables,then the first stage would be as described in the preceding paragraphs:the mth iteration would include:

$\begin{matrix}{r^{{\lbrack 1\rbrack}{(m)}} = {r^{{\lbrack 1\rbrack}{({m - 1})}} - {G_{{{\lbrack 1\rbrack}m} - 1}{\hat{s}}^{{\lbrack 1\rbrack}{({m - 1})}}}}} \\{= {r - {\sum\limits_{1 \leq k \leq {m - 1}}{G_{{\lbrack 1\rbrack}k}{\hat{s}}^{{\lbrack 1\rbrack}{(k)}}}}}}\end{matrix}$F _([1]m)=Λ_(m) H _(m) ^(H) [K _(m)(K _(m)(I−

_([1]m) ²)

_(m) K _(m) ^(H)+σ² I _(Q)]⁻¹x^([1](m))=F_([1]m)r^([1](m)) . . .ŝ^([1](m))=MLD_(m){x^([1](m))}G_([1]m)=H_(m)R_([1]m)Then the second stage would have first stage hard estimates ofinterfering subsets of symbols to include in the cancellations, so

$\begin{matrix}{r^{{\lbrack 2\rbrack}{(m)}} = {r^{{\lbrack 2\rbrack}{({m - 1})}} - {G_{{{\lbrack 2\rbrack}m} - 1}{\hat{s}}^{{\lbrack 2\rbrack}{({m - 1})}}}}} \\{= {r - {\sum\limits_{1 \leq k \leq {m - 1}}{G_{{\lbrack 2\rbrack}k}{\hat{s}}^{{\lbrack 2\rbrack}{(k)}}}} - {\sum\limits_{{m + 1} \leq j \leq M}{G_{{\lbrack 1\rbrack}j}{\hat{s}}^{{\lbrack 1\rbrack}{(j)}}}}}}\end{matrix}$Note that the hard estimates from the first stage for the subset ofsymbols being detected, ŝ^([1](m)), are not part of the cancellation.And the correlation matrices

_([2]m) can use first stage correlations ρ_([1]k) for k in J_(m),whereas R_([2]m) is used after the mth iteration and will havecorrelations ρ_([2]k) for k in J_(m). And so forth for subsequentstages.6. Ordered Detection

Ordered detection based on the symbol post-detectionsignal-to-interference-plus-noise ratio (SINR) is often used to furtherimprove performance. error. Selecting the detection ordering in theforegoing preferred embodiments amounts to selecting the index sets I₁,I₂, . . . , I_(M) that form a partition of {1, 2, . . . , P}. One mayselect a pre-determined detection ordering that does not depend on thechannel realization H. Analogous to the conventional iterativedetection, further improvement can be obtained by selecting the orderingas a function of H based on a selected performance measure as in thefollowing examples.

(1) Pre-ML SINR Measures

At the start of the mth iteration, compare the (Σ_(m≦k≦M)P_(k)) SINRvalues associated with the index set J_(m), and select index set I_(m)to label the P_(m) largest SINR values. Note that closed formexpressions for the SINR as a function of the channel realization can beobtained for different detection criteria. In particular, define theSINR associated with symbol s_(p) at the mth iteration as γ_(p) ^((m)).For the zero-forcing criterion this yields for pεI_(m):γ_(p) ^((m))=λ_(p)/σ²[(K _(m) ^(H) K _(m))⁻¹]_(p,p)Whereas the MMSE criterion together with the feedback error compensationyields for pεI_(m):

γ p ( m ) = λ p ⁢ h p H [ ∑ n ∈ J m + l ⁢ λ n ⁡ ( 1 - ρ n 2 ) ⁢ h n ⁢ h n H +∑ k ∈ I m - p ⁢ λ k ⁢ h k ⁢ h k H + σ 2 ⁢ I Q ] - 1 ⁢ h p = λ p / [ σ 2 ⁢ m -1 ⁢ ( K m H ⁢ K m ) ⁢ ( I - m 2 ) + σ 2 ⁢ m - 1 ) - 1 ] p , p - 1where h_(k) is the kth column of H, and the case of error-free feedbackis a special case.

(2) Post-ML SINR measures

Intuitively, better detection ordering can be obtained when a measure ofSINR after the ML detection. Several possible post-ML detection measurescan be derived from the symbol error rate (SER) expression of MLdetection. Starting from the ML metric above, scale by Ψ_(m) ^(−1/2) todefine:

$\begin{matrix}{y^{(m)}\overset{def}{=}{\Psi_{m}^{{- 1}/2}x^{(m)}}} \\{\Theta_{m}\overset{def}{=}{\Psi_{m}^{{- 1}/2}F_{m}K_{m}}} \\{j_{m}\overset{def}{=}{\Psi_{m}^{{- 1}/2}i_{m}}}\end{matrix}$ Then $\begin{matrix}{y^{(m)} = {\Psi_{m}^{{- 1}/2}x^{(m)}}} \\{= {\Psi_{m}^{{- 1}/2}( {{F_{m}K_{m}s^{(m)}} + i_{m}} )}} \\{= {{\Psi_{m}^{{- 1}/2}\Theta_{m}s^{(m)}} + j_{m}}}\end{matrix}$Here the scaled residual interference-plus-noise term, j_(m), is whitewith E[j_(m)j_(m) ^(H)]=I. Let

_(m) denote the P_(m)-dimensional product of constellations of thesymbols with indices in I_(m); that is,

_(m)=Π_(kεIm) C_(k) where C_(k) is the symbol constellation for s_(k)and the product is over k in I_(m). Also let N_(m) denote the size of

_(m), so N_(m)=Π_(kεIm) |C_(k)|. An upper bound of symbol error rate(SER) corresponding to MLD_(m) can be derived as follows:

SER m = ⁢ Pr ⁡ ( s ^ ( m ) ≠ s ( m ) ) = ⁢ ( 1 / N m ) ⁢ ∑ α ∈ ⁢ m ⁢ Pr ⁡ ( s ^( m ) ≠ s ( m ) | s ( m ) = α ) = ⁢ ( 1 / N m ) ⁢ ∑ α ∈ ⁢ m ⁢ ∑ α . , α . ≠α ⁢ Pr ( s ^ ( m ) = α . | s ( m ) = α ) = ⁢ ( 1 / N m ) ⁢ ∑ α ∈ ⁢ m ⁢ ∑ α ., α . ≠ α ⁢ Pr ( ⋂ β ≠ α . ⁢ {  y ( m ) - Θ m ⁢ α .  2 ≤ ⁢  y ( m ) - Θ m⁢β  2 } ) ≤ ⁢ ( 1 / N m ) ⁢ ∑ α ∈ ⁢ m ⁢ ∑ α . , α . ≠ α ⁢ Pr ⁡ (  y ( m ) - Θm ⁢ α .  2 ≤  y ( m ) - Θ m ⁢ α  2 ) = ⁢ ( 1 / N m ) ⁢ ∑ α ∈ ⁢ m ⁢ ∑ α . ,α . ≠ α ⁢ ℚ ⁡ ( √ (  Θ m ⁡ ( α - α . )  2 / 2 ) )where the

function is the area under the Gaussian tail. This upper bound onSER_(m) can be used to select the detection order: take I_(m) tominimize the upper bound. Thus this measure is post-ML based on the SERupper bound.

A looser upper bound can be derived from the fact that for α,

ε

_(m):∥Θ_(m)(α−

)∥²≧min _(1≦i≦Pm)(∥[Θ_(m)]_(:,i)∥² d ² _(min,i))where d² _(min,i) is the minimum distance among symbols in the symbolconstellation associated with the ith column of Θ_(m), [Θ_(m)]_(:,i).Thus,

SER m ≤ ( 1 / N m ) ⁢ ∑ α ∈ ⁢ m ⁢ ∑ α . , α . ≠ α ⁢ ℚ ⁡ ( √ (  Θ m ⁡ ( α - α. )  2 / 2 ) ) ≤ ( N m - 1 ) ⁢ ℚ ( √ ( min 1 ≤ i ≤ P m ⁢ (  [ Θ m ] : ,i  2 ⁢ d min , i 2 ) ) )When the same constellation (modulation scheme) is used for all of thetransmit antennas, the minimum distance is common and the upper boundsimplifies to:SER _(m)≦(N _(m)−1)

(√(d ² _(min) min_(1≦i≦Pm)(∥[Θ_(m)]_(:,i)∥²) ))This upper bound yields the following post-ML measure for selecting thedetection order: at the mth iteration, select I_(m) to maximizemin_(1≦i≦Pm)(∥[Θ_(m)]_(:,i)∥²). This measure is thus post-ML based onthe channel norm after MMSE transformation.7. Decision Alternatives

The foregoing preferred embodiments hard decision methods, MLD_(m),convert x_(k) ^((m)) into ŝ_(k) ^((m)) use a maximum likelihood approachwhich applies a hard-limiter (hard-clipping) non-linearity. Alternativenon-linearities, generically denoted as ŝ^((m))=

{x^((m))}, such as hyperbolic tangent, may result in better performance.In this case the decision statistics that are close to zero areeffectively scaled instead of clipped; the scaling depends upon thevariance of the residual interference plus noise. For example, for asingle BPSK symbol:

Hyperbolic tangent: ??_(tanh)(z) = tanh (Re{z}/ξ) Soft linear clipped:$\begin{matrix}{{{??}_{SL}(z)} = {+ 1}} & {{{if}\mspace{14mu}{Re}\{ z \}} > \xi} \\{= {- 1}} & {{{if}\mspace{14mu}{Re}\{ z \}} < {- \xi}} \\{= {{Re}\{ z \}}} & {{{if}\mspace{14mu}{{{Re}\{ z \}}}} \leq \xi}\end{matrix}\quad$where ξ is a non-negative constant. For higher order modulation, thedecision is done in the bit level following the bit log likelihood ratiocomputation (as described in Section 8). Alternatively, for QPSK, thereal and imaginary parts of:

(z) can be taken as tan h(Re{z}/ξ) and tan h(lm{z}/ξ), respectively; orthe corresponding soft linear with Re{z} and lm{z}, respectively.Further, for higher order (larger constellations) the soft linearextends by selecting the constant to make

(z)=z for z in the constellation. And for two or higher-dimensionalconstellations, extend by components. For example, with a subset ofsymbols of size P_(m)=2, the ML decision on the soft estimate 2-vectorx^((m)) gives the 2-vector of symbols in the product constellationclosest to x^((m)); whereas, the soft linear would give the 2-vector [

_(SL)(x₁ ^((m))),

_(SL)(x₂ ^((m)))].

While any of those non-linear estimators can be used, the optimal softestimator is the conditional expectation estimator (analogous to Section2), which is

${\hat{s}}^{(m)} = {{E\lbrack s^{(m)} \middle| x^{(m)} \rbrack} = \frac{\sum\limits_{c}^{\;}{c \times {\exp( {- {{\Psi_{m}^{{- 1}/2}( {x^{(m)} - {F_{m}H_{m}c}} )}}^{2}} )}}}{\sum\limits_{c}^{\;}{\exp( {- {{\Psi_{m}^{{- 1}/2}( {x^{(m)} - {F_{m}H_{m}c}} )}}^{2}} )}}}$where the summation is over the set formed by the product of P_(m)constellations of symbols with indices in I_(m) (see discussion at theend of Section 3).

For systems with large constellation and/or a number of antennas, thecomputational burden may be prohibitive as it increases exponentiallywith the number of transmit antennas. In this case, sphere decoding typealgorithm can be used to reduce computational complexity at the expenseof some performance (while retaining most of the gain of maximumlikelihood detector).

8. Decoding

The demodulator converts the soft symbol estimates x⁽¹⁾, x⁽²⁾, . . . ,x^((M)) output by an M-iteration detector into conditionalprobabilities; and the conditional probabilities translate intobit-level log likelihood ratios (LLRs) for (sequence) decoding. In moredetail, the LLRs in terms of the bits u_(pk) which define theconstellation symbols s_(p) (e.g., two bits for a QPSK symbol, four bitsfor a 16QAM symbol, etc.) are defined (dropping the superscripts on thex_(p)) as

$\begin{matrix}{{{LLR}( u_{pk} )} = {\log\{ {{P\lbrack {u_{pk} =  1 \middle| x_{pk} } \rbrack}/{P\lbrack {u_{pk} =  0 \middle| x_{pk} } \rbrack}} \}}} \\{= {{\log\{ {P\lbrack {u_{pk} =  1 \middle| x_{pk} } \rbrack} \}} - {\log\{ {P\lbrack {u_{pk} =  0 \middle| x_{pk} } \rbrack} \}}}}\end{matrix}$The LLRs can be computed using a channel model. For example,

$\begin{matrix}{{{LLR}( u_{pk} )} = {\log\{ {{P\lbrack {u_{pk} =  1 \middle| x_{p} } \rbrack}/{P\lbrack {u_{pk} =  0 \middle| x_{p} } \rbrack}} \}}} \\{{\log\{ {{P( { x_{p} \middle| u_{pk}  = 1} )}/{p( { x_{p} \middle| u_{p\; k}  = 0} )}} \}} +} \\{\log\{ {{P\lbrack {u_{pk} = 1} \rbrack}/{P\lbrack {u_{nk} = 0} \rbrack}} \}}\end{matrix}$where the first log term includes the probability distribution of thedemodulated symbol x_(p) which can be computed using the channel model.The second log term is the log of the ratio of a priori probabilities ofthe bit values and typically equals 0. So for an AWGN channel where theresidual interference (interference which is not cancelled) is also azero-mean, normally-distributed, independent random variable, thechannel model gives:p(x _(p) |s _(p) =c)˜exp(−|x _(p) −c| ²/γ_(p))where c is a symbol in the symbol constellation and γ_(p) is anormalization typically derived from the channel characteristics and thedetector type. Indeed, the following section describes knownnormalizations and preferred embodiment normalizations. Of course, γ_(p)is just twice the variance of the estimation error random variable.

Then compute LLRs by using an approximation which allows directapplication of the channel model. Takep(x_(p)|u_(pk)=1)=p(x_(p)|s_(p)=c_(pk=1)) where C_(pk=1) is the symbolin the sub-constellation of symbols with kth bit equal 1 and which isthe closest to x_(p); that is, c_(pk=1) minimizes |x_(p)−c_(k=1)|² forC_(k=1) a symbol in. the sub-constellation with kth bit equal to 1.Analogously for p(x_(p)|u_(pk)=0) using the sub-constellation of symbolswith kth bit equal 0. Then with equal a priori probabilities of the bitvalues and the notation subscript k=1 and k=0 indicating symbols withkth bit 1 and 0, respectively, the approximation yields

$\begin{matrix}{{{LLR}( u_{pk} )} = {\log\;\{ {{p( { x_{p} \middle| u_{pk}  = 1} )}/{p( { x_{p} \middle| u_{pk}  = 0} )}} \}}} \\{\cong {{1/\gamma_{p}} \cdot \{ {{\min_{k = 0}{{x_{p} - c_{k = 0}}}^{2}} - {\min_{k = 1}{{x_{p} - c_{k = 1}}}^{2}}} \}}}\end{matrix}$Thus the LLR computation just searches over the two symbolsub-constellations for the minima.

The LLRs are used in decoders for error correcting codes such as Turbocodes (e.g., iterative interleaved MAP decoders with BCJR or SOVAalgorithm using LLRs for each MAP) and convolutional codes (e.g. Viterbidecoders). Such decoders require soft bit statistics (in terms of LLR)from the detector to achieve their maximum performance (hard bitstatistics with Hamming instead of Euclidean metrics can also be usedbut result in approximately 3 dB loss). Alternatively, direct symboldecoding with LLRs as the conditional probability minus the a prioriprobability could be used. FIGS. 4 a-4 b illustrate the 3GPP Turboencoder and an iterative decoder which includes two MAP blocks, aninterleaver, a de-interleaver, and feedback for iterations.

9. Simulations

Simulations for 4×4 MIMO QPSK and 16QAM systems with P=Q=4 in spatiallyuncorrelated (IID) and correlated MIMO channels were performed tocompare a preferred embodiment with other detection methods. Thecorrelated spatial channel profile has the average and peak crosscorrelation of 0.16 and 0.40, respectively. FIGS. 3 a-3 d comparesimulations with the following six detection methods including fourpreferred embodiment iterative reduced state ML variations:

(1) IMMSE UNORD: regular iterative MMSE without detection ordering.

(2) IMMSE ORD: regular iterative MMSE with detection ordering based onpost-detection SINR.

(3) IRSML UNORD: MMSE-based iterative reduced state ML for M=2, P₁=P₂=2and without ordering.

(4) IRSML ORD Pre-ML SINR: MMSE-based iterative reduced state ML forM=2, P₁=P₂=2 and with ordering based on SINR after MMSE transformation.

(5) IRSML ORD Post-ML Norm: MMSE-based iterative reduced state ML forM=2, P₁=P₂=2 and with ordering based on channel norm after MMSEtransformation.

(6) IRSML ORD Post-ML SER: MMSE-based iterative reduced state ML forM=2, P₁=P₂=2 and with ordering based on SER bound after ML detection.

The raw bit error rate (BER) versus signal-to-noise ratio (E_(b)/N₀)results for QPSK in IID and correlated channels appear in FIGS. 3 a-3 b,respectively, and for 16QAM in IID and correlated channels appear inFIGS. 3 c-3 d, respectively. Observe that the gain of IRSMLover regularIMMSE is more dramatic in spatially correlated channels and when nodetection ordering is used. Notice also that IRSML with SER-baseddetection ordering results in a gain of 10-dB for 16QAM in correlatedchannels at BER=10⁻³ compared to the regular IMMSE with detectionordering.

The computational complexity of preferred embodiment iterative reducedstate ML methods can be compared to the full ML and the regular MMSEmethods. In particular, consider two aspects: matrix inversioncomputation and hard decision estimation. For comparison, assume thatthe preferred embodiments use equal size subsets (P_(m)=P/M) and allmethods use the same constellation (size C) for all symbols. Then thefollowing table indicates the complexity of the methods.

Detector Matrix inversion Hard decision Regular iterative ~PxP³ ~PxCMaximum likelihood none ~C^(P) IRSML ~MxP³ ~MxC^(P/M)

Thus for the preferred embodiments with two subsets (M=P/2 and P_(m)=2),the computational complexity of the preferred embodiments is comparableto the regular iterative detector for small to moderate constellationsize. For large constellation size (such as 64-QAM or 256-QAM), thecomplexity increase for the preferred embodiments over the regulariterative detector becomes significant. However, its performance gain isexpected to become more dramatic as indicated by the simulations.

1. A method of detection in a multiple-input, multiple-output wirelesscommunication system, comprising the steps of: (a) receiving a signalrepresenting a set of P symbols, one symbol transmitted from each of Pantennas where P is a positive integer greater than 2; (b) jointlyestimating a subset of P₁ symbols of said set of P symbols where P₁ is apositive integer; (c) after step (b), jointly estimating a subset of P₂symbols of said set of P symbols where P₂ is a positive integer andwherein said subset of P₁ symbols and said subset of P₂ symbols aremembers of a partition of said set of P symbols and P₁+P₂ is greaterthan 2 and wherein P₁=P₂=P/2 when there are 2 antennas.
 2. The method ofclaim 1, further comprising: (a) after step (c) of claim 1, for each min the set {3, . . . , M}, jointly estimating a subset of P_(m) symbolsof said set of P symbols where P_(m) is a positive integer and whereinsaid subset of P_(m) symbols is a member of a partition of said set of Psymbols and P₁+P₂+ . . . +P_(M)=P where M is a positive integer.
 3. Themethod of claim 2, wherein: (a)P₁=P₂= . . . =P_(M)=P/M.
 4. The method ofclaim 1, wherein: (a) said jointly estimating of step (b) of claim 1includes a decision using P₁-vector of soft estimates F₁ r where r is aQ-vector of said received signals of step (a) of claim 1 and F₁ is aP₁×Q matrix for zero-forcing estimation; (b) said jointly estimating ofstep (C) of claim 1 includes a decision using P₂-vector of softestimates F₂(r−G₁s⁽¹⁾) where F₂ is a P₂×Q matrix for zero-forcingestimation, G₁ is a Q×P₁ matrix for zero-forcing feedback cancellation,and s⁽¹⁾ is the P₁-vector estimation result of step (b) of claim
 1. 5.The method of claim 1, wherein: (a) said jointly estimating of step (b)of claim 1 includes a decision using P₁-vector of soft estimates F₁ rwhere r is a Q-vector of said received signals of step (a) of claim 1and F₁ is a P₁×Q matrix for minimum mean square error estimation (b)said jointly estimating of step (c) of claim 1 includes a decision usingP₂-vector of soft estimates F₂(r−G₁s⁽¹⁾) where F₂ is a P₂×Q matrix forminimum mean square error estimation, G₁ is a Q×P₁ matrix forzero-forcing feedback cancellation, and s⁽¹⁾ is the P₁-vector estimationresult of step (b) of claim
 1. 6. The method of claim 1, wherein: (a)said jointly estimating of step (b) of claim 1 includes a decision usingP₁-vector of soft estimates F₁ r where r is a Q-vector of said receivedsignals of step (a) of claim 1 and F₁ is a P₁×Q matrix for minimum meansquare error estimation (b) said jointly estimating of step (c) of claim1 includes a decision using P₂-vector of soft estimates F₂(r−G₁s⁽¹⁾)where F₂ is a P₂×Q matrix for minimum mean square error estimationincluding feedback error compensation, G₁ is a Q×P₁ matrix forzero-forcing feedback cancellation including feedback errorcompensation, and s⁽¹⁾ is the P₁-vector estimation result of step (b) ofclaim
 1. 7. The method of claim 1, wherein: (a) said subset of P₁symbols of step (b) of claim 1 is determined according tosignal-to-interference-plus-noise ratios of said P symbols prior to adecision in said estimating.
 8. The method of claim 1, wherein: (a) saidsubset of P₁ symbols of step (b) of claim 1 is determined according toprojected signal-to-interference-plus-noise ratios of said P symbolsafter a decision in said estimating.
 9. The method of claim 1, wherein:(a) said jointly estimating of step (b) of claim 1 includes a maximumlikelihood decision; and (b) said jointly estimating of step (c) ofclaim 1 includes a maximum likelihood decision.
 10. The method of claim1, wherein: (a) said jointly estimating of step (b) of claim 1 includesa soft decision; and (b) said jointly estimating of step (c) of claim 1includes a soft decision.
 11. The method of claim 1, further comprising:(a) jointly re-estimating said subset of P₁ symbols using errorcompensation determined by said jointly estimating said subset of P₂symbols of step (c) of claim 1.